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Intermediate methods in observational epidemiology 2008 Instructor: Moyses Szklo

Intermediate methods in observational epidemiology 2008 Instructor: Moyses Szklo. Measures of Disease Frequency. MEASURES OF RISK. Absolute measures of event (including disease) frequency : Incidence and Incidence Odds Prevalence and Prevalence Odds.

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Intermediate methods in observational epidemiology 2008 Instructor: Moyses Szklo

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  1. Intermediate methods in observational epidemiology 2008Instructor: Moyses Szklo Measures of Disease Frequency

  2. MEASURES OF RISK • Absolute measures of event (including disease) frequency: • Incidence and Incidence Odds • Prevalence and Prevalence Odds

  3. What is "incidence"?Two major ways to define incidence • Cumulative incidence (probability) SURVIVAL ANALYSIS (Unit of analysis: individual) • Rate or Density ANALYSIS BASED ON PERSON-TIME (Unit of analysis: time)

  4. 1.0 Survival Time • OBJECTIVE OF SURVIVAL ANALYSIS: To compare the “cumulative incidence” of an event (or the proportion surviving event-free) in exposed and unexposed (characteristic present or absent) while adjusting for time to event (follow-up time) • BASIS FOR THE ANALYSIS • NUMBER of EVENTS • TIME of occurrence

  5. Need to precisely define: • “EVENT” (failure): • Death • Disease (diagnosis, start of symptoms, relapse) • Quit smoking • Menopause • “TIME”: • Time from recruitment into the study • Time from employment • Time from diagnosis (prognostic studies) • Time from infection • Calendar time • Age

  6. Example: • Follow up of 6 patients (2 yrs) • 3 Deaths • 2 censored (lost) before 2 years • 1 survived 2 years Question: What is the Cumulative Incidence (or the Cumulative Survival) up to 2 years?

  7. Person ID (24) 1 (6) 2 (18) 3 (15) 4 (13) 5 (3) 6 Jan 2000 Jan 2001 Jan 1999 Crude Survival: 3/6= 50% Death Censored observation (lost to follow-up, withdrawal) ( ) Number of months to follow-up

  8. (6) (18) (15) (3) Change time scale to “follow-up” time: Person ID (24) 1 2 3 4 (13) 5 6 1 2 0 Follow-up time (years)

  9. Cumulative Survival 100 82 43 Follow-up time Year 1 Year 2 One solution: • Actuarial life table Assume that censored observations over the period contribute one-half the persons at risk in the denominator (censored observations occur uniformly throughout follow-up interval). ID 1 (24) 2 (6) 3 (18) 4 (15) 5 (13) 6 (3) 1 2 0 Follow-up time (years) It can be also calculated for years 1 and 2 separately: Year 1: S(Y1)= [1 - {1 ÷ [6 – ½(1)]}= 0.82 Year 2: S(Y2)= [1 – {2 ÷ [4 – ½(1)]}= 0.43 S(2yrs)= 0.82 × 0.43= 0.35

  10. (6) (18) (15) (3) KAPLAN-MEIER METHODE.L. Kaplan and P. Meier, 1958* Calculate the cumulative probability of event (and survival) based on conditional probabilities at each event time Step 1: Sort the survival times from shortest to longest Person ID (24) 1 2 3 4 (13) 5 6 1 2 0 Follow-up time (years) *Kaplan EL, Meier P.Nonparametric estimation from incomplete observations. J Am Stat Assoc 1958;53:457-81.

  11. (6) (18) (15) (24) 1 (3) 2 3 6 KAPLAN-MEIER METHODE.L. Kaplan and P. Meier, 1958* Calculate the cumulative probability of event (and survival) based on conditional probabilities at each event time Step 1: Sort the survival times from shortest to longest Person ID (13) 5 4 1 2 0 Follow-up time (years) *Kaplan EL, Meier P.Nonparametric estimation from incomplete observations. J Am Stat Assoc 1958;53:457-81.

  12. (6) (13) (18) (15) (24) 1 (3) 2 3 6 Step 2: For each time of occurrence of an event, compute the conditional survival Person ID 5 4 1 2 0 Follow-up time (years) When the first event occurs (3 months after beginning of follow-up), there are 6 persons at risk. One dies at that point; 5 of the 6 survive beyond that point. Thus: • Incidence of event at exact time 3 months: 1/6 • Probability of survival beyond 3 months: 5/6

  13. (13) (18) (15) (24) 1 3 Person ID (3) 6 (6) 2 5 4 1 2 0 Follow-up time (years) When the second event occurs (13 months), there are 4 persons at risk. One of them dies at that point; 3 of the 4 survive beyond that point. Thus: • Incidence of event at exact time 13 months: 1/4 • Probability of survival beyond 13 months: ¾

  14. (18) (24) 1 3 Person ID (3) 6 (6) 2 (13) 5 (15) 4 1 2 0 Follow-up time (years) When the third event occurs (18 months), there are 2 persons at risk. One of them dies at that point; 1 of the 2 survive beyond that point. Thus: • Incidence of event at exact time 18 months: 1/2 • Probability of survival beyond 18 months: 1/2

  15. CONDITIONAL PROBABILITY OF AN EVENT (or of survival) The probability of an event (or of survival) at time t (for the individuals at risk at time t), that is, conditioned on being at risk at exact time t. Step 3: For each time of occurrence of an event, compute the cumulativesurvival (survival function), multiplying conditional probabilities of survival. 3 months: S(3)=5/6=0.833 12 months: S(13)=5/63/4=0.625 18 months: S(18)=5/6 3/41/2 =0.3125

  16. Time (mo) 0.833 3 0.625 13 0.3125 18 0.833 Survival 0.625 1.00 0.80 0.3125 0.3125 0.60 0.40 0.20 0 10 25 5 20 15 Month of follow-up Plotting the survival function: Si The cumulative incidence (up to 24 months): 1-0.3125 = 0.6875 (or 69%)

  17. Time (mo) 0.833 3 0.625 13 0.3125 18 Cumulative Survival 1.00 0.8 0.80 0.6 0.60 0.3 0.40 0.20 0 10 25 5 20 15 Plotting the survival function: Month of follow-up

  18. EXPERIMENTAL STUDY Placebo CEE CEE Placebo Cumulative Hazards for Coronary Heart Disease and Stroke in the Women’s Health Initiative Randomized Controlled Trial (The WHI Steering Committee. JAMA 2004;291:1701-1712)

  19. Time (mo) 0.833 3 0.625 13 0.3125 18 Cumulative Survival Cumulative Hazard 1.00 1.00 0.8 0.7 0.80 0.80 0.6 0.60 0.60 0.3 0.40 0.40 0.2 0.4 0.20 0.20 0 10 25 5 20 15 Plotting the survival function: Month of follow-up The cumulative incidence (hazard) at the end of 24 months: 1-0.3 = 0.7 (or 70%)

  20. ACTUARIAL LIFE TABLE VS KAPLAN-MEIER If N is large and/or if life-table intervals are small, results are similar • Survival after diagnosis of Ewing’s sarcoma

  21. ASSUMPTIONS IN KAPLAN-MEIER SURVIVAL ESTIMATES • (If individuals are recruited over a long period of time) No secular trends Follow-up time Calendar time

  22. ASSUMPTIONS IN SURVIVAL ESTIMATES (Cont’d) • Censoring is independent of survival (uninformative censoring): Those censored at time t have the same prognosis as those remaining. Types of censoring: • Lost to follow-up • Migration • Refusal • Death (from another cause) • Administrative withdrawal (study finished)

  23. Calculation of incidenceStrategy #2ANALYSIS BASED ON PERSON-TIME CALCULATION OF PERSON-TIME AND INCIDENCE RATES (Unit of analysis: time) Example 1 Observe 1st graders, total 500 hours Observe 12 accidents Accident rate: IT IS NOT KNOWN WHETHER 500 CHILDREN WERE OBSERVED FOR 1 HOUR, OR 250 CHILDREN OBSERVED FOR 2 HOURS, OR 100 CHILDREN OBSERVED FOR 5 HOURS… ETC.

  24. It is also possible to calculate the incidence rates per person-year separately for shorter periods during the follow-up: For year 1: For year 2: Step 2: Calculate rate per person-year for the total follow-up period: CALCULATION OF PERSON-TIME AND INCIDENCE RATES Example 2 Person ID (3) 6 (6) 2 (13) Step 1: Calculate denominator, i.e. units of time (years) contributed by each individual, and total: 5 (15) 4 (18) 3 (24) 1 1 2 0 Follow-up time (years)

  25. Notes: • Rates have units (time-1). • Proportions (e.g., cumulative incidence) are unitless. • As velocity, rate is an instantaneous concept. The choice of time unit used to express it is totally arbitrary. E.g.: 0.024 per person-hour = 0.576 per person-day = 210.2 per person-year 0.46 per person-year = 4.6 per person-decade

  26. No. of person-years of follow-up Death rate per person-time (person-year) 5 deaths/25.0 person-years= 0.20 or 20 deaths per 100 person-years Death rate per average population, estimated at mid-point of follow-up Mid-point (median) population (When calculating yearly rate in Vital Statistics) = 12.5 Death rate= 5/12.5 per 2 years= 0.40 Average annual death rate= 0.40/2= 0.20 or 20/100 population D, deaths C, censored

  27. No. of person-years of follow-up Death rate per person-time (person-year) 5 deaths/25.0 person-years= 0.20 or 20 deaths/100 person-years Death rate per average population, estimated at mid-point of follow-up Mid-point (median) population (When calculating yearly rate in Vital Statistics) = 12.5 Death rate= 5/12.5 per 2 years= 0.40 Average annual death rate= 0.40/2= 0.20 or 20/100 population D, deaths C, censored

  28. Notes: Rates have an undesirable statistical property • Rates can be more than 1.0 (100%): • 1 person dies exactly after 6 months: • No. of person-years: 1 x 0.5 years= 0.5 person-years

  29. No. PY PRE meno No. PY POST meno 7 3 10 6 2 9 5 ID 1 8 4 C 5 0  0 6  1 0 C 5 5 3 3  18 17 : Myocardial Infarction; C: censored observation. Use of person-time to account for changes in exposure status (Time-dependent exposures) Example: Adjusting for age, are women after menopause at a higher risk for myocardial infarction? Year of follow-up 4 3 Note: Event is assigned to exposure status when it occurs Rates per person-year: Pre-menopausal = 1/17 = 0.06 (6 per 100 py) Post-menopausal = 2/18 = 0.11 (11 per 100 py) Rate ratio = 0.11/0.06 = 1.85

  30. ASSUMPTIONS IN PERSON-TIME ESTIMATES Risk is constant within each interval for which person-time units are estimated (no cumulative effect): • N individuals followed for t time  t individuals followed for N time • However, are 10 smokers followed for 1 year comparable to 1 smoker followed for 10 years (both: 10 person-years) • No secular trends (if individuals are recruited over a relatively long time interval) • Losses are independent from survival Rate for 1st Year= 0.21/PY Rate for 2nd Year= 1.09/ PY Total for 2 years = 0.46/PY

  31. ASSUMPTIONS IN PERSON-TIME ESTIMATES Risk is constant within each interval/period for which person-time units are estimated (no cumulative effect): • N individuals followed for t time  t individuals followed for N time • However, are 10 smokers followed for 1 year comparable to 1 smoker followed for 10 years (both: 10 person-years) • No secular trends (if individuals are recruited over a relatively long time interval) • Losses are independent of survival

  32. SUMMARY OF ESTIMATES

  33. POINT REVALENCE

  34. Point Prevalence“The number of affected persons present at the population at a specific time divided by the number of persons in the population at that time”Gordis, 2000, p.33 Relation with incidence --- Usual formula: Point Prevalence = Incidence x Duration* P = I x D True formula: * Average duration (survival) after disease onset.

  35. ODDS

  36. OddsThe ratio of the probabilities of an event to that of the non-event. Example: The probability of an event (e.g., death, disease, recovery, etc.) is 0.20, and thus the odds is: That is, for every person with the event, there are 4 persons without the event.

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